Should I prove this by contradiction then? If the Galois group is not the cyclic group of order 3, then some roots are complex?
Or assume that there is a root that's not real, then the Galois group is not of order 3...

Yes. If there is a complex root then there is another complex root that is its conjugate. If you let be complex conjugation then is an automorphism of order 2. Which contradicts that the Galois group is cyclic of order 3 (therefore all its automorphism must have order dividing 3).